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Phase transition in the family of p-resistances

2011

Conference Paper

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We study the family of p-resistances on graphs for p ≥ 1. This family generalizes the standard resistance distance. We prove that for any fixed graph, for p=1, the p-resistance coincides with the shortest path distance, for p=2 it coincides with the standard resistance distance, and for p → ∞ it converges to the inverse of the minimal s-t-cut in the graph. Secondly, we consider the special case of random geometric graphs (such as k-nearest neighbor graphs) when the number n of vertices in the graph tends to infinity. We prove that an interesting phase-transition takes place. There exist two critical thresholds p^* and p^** such that if p < p^*, then the p-resistance depends on meaningful global properties of the graph, whereas if p > p^**, it only depends on trivial local quantities and does not convey any useful information. We can explicitly compute the critical values: p^* = 1 + 1/(d-1) and p^** = 1 + 1/(d-2) where d is the dimension of the underlying space (we believe that the fact that there is a small gap between p^* and p^** is an artifact of our proofs. We also relate our findings to Laplacian regularization and suggest to use q-Laplacians as regularizers, where q satisfies 1/p^* + 1/q = 1.

Author(s): Alamgir, M. and von Luxburg, U.
Book Title: Advances in Neural Information Processing Systems 24
Pages: 379-387
Year: 2011
Day: 0
Editors: J Shawe-Taylor and RS Zemel and P Bartlett and F Pereira and KQ Weinberger

Department(s): Empirical Inference
Research Project(s): Random geometric graphs
Bibtex Type: Conference Paper (inproceedings)

Event Name: Twenty-Fifth Annual Conference on Neural Information Processing Systems (NIPS 2011)
Event Place: Granada, Spain

Digital: 0

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BibTex

@inproceedings{Alamgirv2011,
  title = {Phase transition in the family of p-resistances},
  author = {Alamgir, M. and von Luxburg, U.},
  booktitle = {Advances in Neural Information Processing Systems 24},
  pages = {379-387},
  editors = {J Shawe-Taylor and RS Zemel and P Bartlett and F Pereira and KQ Weinberger},
  year = {2011}
}